Abstract: Ordinal covariates often serve as inexpensive, coarsened or noisy measurements of continuous inputs. Many mixed-input Gaussian process (GP) models assign a single latent location or covariance structure to each ordinal level, leaving variation among individuals at the same level unmodeled. We propose an errors-in-variables GP (EIV-GP) for settings where quantitative inputs, ordinal proxies, and responses are widely available, but the underlying continuous inputs are measured only in a calibration subset. The model places a GP prior on the response surface $f(x,u)$ and relates each individual's continuous input $u$ to the ordinal measurements $c$. Inference averages over the conditional distribution of $u$ given $(x,c)$, accounting for uncertainty in the unmeasured inputs. This allows estimation of both the observable mean surface $m(x,c)$ and the calibrated response surface $f(x,u)$, prediction from ordinal or continuous inputs, and imputation of unmeasured continuous inputs. We show that averaging over latent inputs yields a valid mixed-input GP kernel and recovers deterministic embeddings as point-mass limits. Individual predictive distributions, however, are generally non-Gaussian mixtures with input-dependent variances. We also establish what can be identified without calibration and explain how calibration anchors the latent coordinates and supports recovery of the response surface. Posterior computation integrates out the GP evaluations and a common variance scale, with blocked updates for latent inputs and measurement parameters and a finite prior on kernel length scales for the latent inputs. We describe numerical studies comparing EIV--GP with existing mixed-input GPs in prediction, surface estimation, and recovery of individual latent inputs.
Bio: Dr. Sheng Jiang is an Assistant Professor in the School of Data Science at The Chinese University of Hong Kong, Shenzhen. His research focuses on Bayesian nonparametrics and computational Bayesian statistics, developing flexible, scalable methods for complex datasets, with applications in traffic flow networks and cryo-electron microscopy. His work on Bayesian nonparametric methods with Gaussian process priors has appeared in The Annals of Statistics and the Journal of the American Statistical Association. Before joining CUHK-Shenzhen, he was a Visiting Assistant Professor at the University of California, Santa Cruz. He earned his Ph.D. in Statistical Science from Duke University and then continued at Duke as a postdoctoral researcher.
